We address the energetic stability of the graphene/SiC(0001) interface and the associated binding mechanism by studying a series of low-strain commensurate interface structures within a density functional scheme. Among the structures with negligible strain, the 6 root 3x6 root 3R30 degrees SiC periodicity shows the lowest interface energy, providing a rationale for its frequent experimental observation. The interface stability is driven by the enhanced local reactivity of the substrate-bonded graphene atoms undergoing sp(2)-to-sp(3) rehybridization (pyramidalization). By this mechanism, relaxed structures of higher stability exhibit more pronounced graphene corrugations at the atomic scale.